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An Introduction to Lie Algebras and the Theorem of Ado

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An Introduction to Lie Algebras and the Theorem of Ado An introduction to Lie algebras and the theorem of Ado Hofer Joachim 28.01.2012 An introduction to Lie algebras and the theorem of Ado Contents Introduction 2 1 Lie algebras 3 1.1 Subalgebras, ideals, quotients . 4 1.2 Nilpotent, solvable, simple and semisimple Lie algebras . 5 2 The representation theory of Lie algebras 7 2.1 Examples . 7 2.2 Modules, submodules, quotient modules . 7 2.3 Structure theorems: Lie and Engel . 8 3 The theorem of Ado for nilpotent Lie algebras. How can a faithful representation be constructed? 11 3.1 Ad hoc examples . 11 3.2 The universal enveloping algebra . 11 3.2.1 Tensor products and the tensor algebra . 11 3.2.2 The universal enveloping algebra of a Lie algebra . 12 3.2.3 The Poincare-Birkhoff-Witt theorem . 12 3.3 Constructing a faithful representation of h1 ............................... 12 3.4 Ado’s theorem for nilpotent Lie algebras . 13 3.5 Constructing a faithful representation for the standard filiform Lie algebra of dimension 4 . 14 3.6 Constructing a faithful representation for an abelian Lie algebra . 15 4 The theorem of Ado 16 4.1 Derivations . 16 4.2 Direct and semidirect sums of Lie algebras . 16 4.3 Proof of Ado’s theorem . 17 4.4 Proof of Neretin’s lemma . 18 4.5 Constructing a faithful representation of the 2-dimensional upper triangular matrices . 19 4.6 Constructing a faithful representation of an abstract Lie algebra . 19 1 Hofer Joachim An introduction to Lie algebras and the theorem of Ado Introduction Lie groups and Lie algebras are of great importance in modern physics, particularly in the context of (continu- ous) symmetry transformations. The Lie algebra of a Lie group is defined as the tangent space to the neutral element of the group and its elements can be seen as "infinitesimal transformations". The Lie algebra of a Lie group is uniquely determined (the converse is not true unless the group is simply connected) and many questions about the group can be reduced to questions about the Lie algebra, which are usually easier to handle. It is particularly pleasant if the algebra can be represented by matrices and an important result in this area is given by Ado’s theorem, which states that any finite-dimensional Lie algebra can be represented by (finite) matrices. In this thesis we will prove Ado’s theorem for nilpotent Lie algebras and provide a method to construct such matrix representations. It is also worthwhile to mention that, although Lie algebras historically arose as a means to study Lie groups, they are meanwhile often studied in their own right. The first chapter contains the basic definitions and some helpful examples. In the second chapter a short introduction to representation theory is given as well as the proofs to Engel’s theorem and Lie’s theorem. The third chapter is reserved for the proof of Ado’s theorem for nilpotent Lie algebras and the theory needed for it (also, the explicit construction of faithful representations is shown in two examples). Finally, the fourth chapter contains the proof to Ado’s theorem for arbitrary Lie algebras as well as the needed theory. The proofs to Engel’s and Lie’s theorems are, for the most part, based on the proofs given in [1]. The proof of Ado’s theorem for nilpotent Lie algebras (section 3) is the same as given in [3], the proof of Ado’s theorem for arbitrary Lie algebras is based on the one given in [5]. In the following, with the exception of the construction of the universal enveloping algebra in chapter 3.2., all vector spaces are assumed to be finite-dimensional. Furthermore, unless mentioned otherwise, the under- lying fields are of characteristic zero and algebraically closed. Note however that Ado’s theorem is valid for Lie algebras over fields of arbitrary characteristic (and, in this context, is sometimes called Iwasawa’s theorem). 2 Hofer Joachim An introduction to Lie algebras and the theorem of Ado 1 Lie algebras Definition. A Lie algebra over a field F is a vector space L over F, together with an operation L × L ! L, (x; y) ! [x; y], which fulfills the following axioms: (i) The operation [:; :] is bilinear. (ii) 8x 2 L :[x; x] = 0. (iii) 8x; y; z 2 L :[x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0. The operation [:; :] is usually called the bracket operation. Property (iii) is called the Jacobi identity. Note that (ii) implies (ii 0) 8x; y 2 L :[x; y] = −[y; x]. If charF 6= 2, (ii) and (ii 0) are equivalent. An important example for a Lie agebra is the set of all linear transformation V ! V (where V is any vec- tor space of a field F), denoted by EndV or, in the context of Lie algebras, gl(V ). gl(V ) is itself a vector space with dimension (dimV )2 and a ring w.r.t. the composition of maps. The bracket operation is defined by [x; y] = xy − yx 8x; y 2 gl(V ), where xy is the composition of x and y. After a basis of V has been chosen, the elements of gl(V ) can be represented as n × n matrices and we write gln(F). The standard basis consists of matrices eij, having a one in the (i; j) position and zeros everywhere else. The Lie bracket is then given by [eij; ekl] = δjkeil − δilejk. Every vector space V with the bracket operation defined as [x; y] = 0 8x; y 2 V is a Lie algebra. Such Lie algebras are called abelian. If L is one-dimensional, it has exactly one basis vector x with the commutation relation [x; x] = 0, so any one-dimensional Lie algebra is abelien. If L is two-dimensional with basis vectors x, y, it is either abelien or [x; y] = αx + βy. Now define a new basis by x0 = αx + βy and take y0 to be an orthogonal vector. It follows that [x0; y0] = γx0 and by scaling y0 ! γ−1y0 we get [x0; y0] = x0. Therefore, up to isomorphism, there are exactly two two-dimensional Lie algebras, one of which is abelian. The Lie algebra sl2(R) is the vector space of all real 2 × 2-matrices with trace zero with the commutator as Lie bracket. It is spanned by the matrices 0 1 0 0 1 0 e = ; f = ; g = . 0 0 1 0 0 −1 The commutation relations are [e; f] = g; [e; g] = −2e; [f; g] = 2f. More generally, sln(R)/sln(C) is the space of all real/complex n × n-matrices with trace zero. The Heisenberg Lie algebra hn is a (2n+1)-dimensional real vector space with a basis fx1; :::; xn; y1; :::; yn; zg and the Lie bracket defined by [xi; yi] = z and all other brackets equal to zero. For example, h1 can be identified with (i.e. "a faithful representation of h1 is given by", see the definitions below) the space of real matrices spanned by 3 Hofer Joachim An introduction to Lie algebras and the theorem of Ado 00 1 01 00 0 01 00 0 11 x = @0 0 0A ; y = @0 0 1A ; z = @0 0 0A 0 0 0 0 0 0 0 0 0 and the Lie bracket is now given by the commutator. The following example shows the connection between a Lie algebra and the corresponding Lie group (by a naive approach). Let Aff(R) be the group of all invertible affine transformations of the line, i.e. Aff(R) = fLa;b : R ! Rj La;bx = ax + b; a; b 2 R; a 6= 0g. The elements of Aff(R) can be written as matrices of the form a b L = , a 6= 0 a;b 0 1 x and the transformation is given by their action on vectors of the form . Now let l be a 2 × 2-matrix, such 1 that 1 + l 2 Aff(R) (where 1 denotes the identity matrix and 2 R). Obviously l has to be of the form a b l = . 0 0 These elements form the Lie algebra aff(R) (with the commutator as the Lie bracket). Note that in general one has to use the exponential map to get from the Lie algebra to the Lie group (and to define the Lie algebra of the Lie group). Another downside of this approach is that it doesn’t explain how the group multiplication leads to the Lie bracket. The affine group Aff(Rn) is the group of all invertible affine transformations Rn ! Rn. As above, it is formed by elements of the form A b L = , A 2 GL ( ), b 2 n A;b 0 1 n R R and the corresponding Lie algebra is given by A b aff( n) = f j A 2 gl ( ); b 2 ng. R 0 0 n R R Definition. A Lie algebra homomorphism between two Lie algebras L1 and L2 is a linear map φ : L1 ! L2, such that 8x; y 2 L1 : φ([x; y]) = [φ(x); φ(y)]. 1.1 Subalgebras, ideals, quotients Definition. A subspace K of a Lie algebra L is called a Lie subalgebra, if x; y 2 K =) [x; y] 2 K. Obviously, every one-dimensional subspace of a Lie algebra is a subalgebra. The Lie algebra sl2(C) is a subalgebra of gl2(C). The normalizer of a subspace K of L is defined as NL(K) = fx 2 Lj[x; K] ⊂ Kg. NL(K) is a subalgebra of L, because for x 2 K and x1; x2 2 NL(K), the Jacobi identity implies [x1; x2]; x = x1; [x2; x] − x2; [x1; x] and therefore [x1; x2] 2 NL(K).
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